The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 15 Sampling Distributions for Sample Means 6 2 10 Lecture 15 1 Review Sampling distribution for a sample count Binomial experiments Binomial distribution Normal approximation Continuity correction Sampling distribution for a sample proportion Exact calculation via binomial distribution Normal approximation No continuity correction How about a sample mean 6 2 10 Lecture 15 2 Diversification A basic principle of investment is that diversification reduces risk That is buying several securities for example stocks rather than just one reduces the variability of the return on an investment The following figures show two distributions of returns in 1987 6 2 10 Lecture 15 3 Distribution of returns for all 1815 stocks on the NYSE for the entire year 1987 The mean return was 3 5 and the distribution shows a very wide spread 6 2 10 Lecture 15 4 Distribution of returns for all possible portfolios that invested equal amounts in each of 5 stocks in 1987 The mean is still 3 5 but the variability is much less 6 2 10 Lecture 15 5 Averages vs Individuals The investment example shows that Averages are less variable than individual observations Averages are closer to normal than individual observations Why 6 2 10 Lecture 15 6 The sampling distribution of X for samples of size 10 compared with the distribution of a single observation 6 2 10 Lecture 15 7 6 2 10 Lecture 15 8 Sampling Distribution of 6 2 10 Lecture 15 X 9 The distributions of X for a 1 obs b 2 obs c 10 obs d 25 obs correction for skewness 6 2 10 Lecture 15 10 Soda Pop The amount of soda pop in each bottle is normally distributed with a mean of 32 2 ounces and a standard deviation of 0 3 ounces Find the probability that a bottle bought by a customer will contain more than 32 ounces Answer Let X denote the amount of soda in a bottle X 32 2 32 32 2 P X 32 P 0 3 0 3 P z 67 0 7486 6 2 10 Lecture 15 x 32 m 32 2 11 Soda Pop Find the probability that a box of four bottles will have a mean of more than 32 ounces of soda per bottle Answer The random variable here is the mean amount of soda per bottle which is normally distributed with a mean of 32 2 and standard deviation of 0 3 2 0 15 Hence X 32 2 32 32 2 P X 32 P 0 15 0 15 P Z 1 33 0 9082 6 2 10 Lecture 15 12 Weekly Income The average weekly income of graduates one year after graduation is 600 Suppose the distribution of weekly income has a standard deviation SD of 100 What is the probability that 25 randomly selected graduates have an average weekly income of less than 550 Answer According to CLT X approximately has a normal distribution with mean 600 and SD 100 5 20 X 600 550 600 P X 550 P 20 20 P Z 2 5 0 0062 6 2 10 Lecture 15 13 Weekly Income continued Suppose we actually found a random sample of 25 graduates with an average weekly income of 550 What can you say about the validity of the claim that the average weekly income is 600 Answer If the population mean is 600 then the probability to have observed a sample mean of 550 is very low 0 0062 The evidence provided by the sample suggests that the assumed average weekly income 600 is unjustified It s more reasonable to believe that the population mean is actually smaller than 600 Then a sample mean of 550 becomes more probable 6 2 10 Lecture 15 14 Sum of independent normal random variables Fact If X Y are independent normal random variables and a b are constants then aX bY also follows a normal distribution with mean E aX bY a E X b E Y and variance 2 2 2 2 2 aX bY a X b Y Use this fact to solve Problem 5 60 c d on page 349 6 2 10 Lecture 15 15 Take Home Message Sampling distribution for a sample mean Mean for a sample mean Variance for a sample mean Central Limit Theorem Normal distribution calculation 6 2 10 Lecture 15 16
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